Elliptic Curves and Elliptic Functions

For a quick definition of many of the terms used here, you may refer to the Glossary.

Contents:

What is an elliptic curve?

An elliptic curve is not an ellipse! The reason for the name is a little more indirect. It has to do, as we shall explain shortly, with "elliptic integrals", which arise in computing the arc length of an ellipse. But this happenstance of nomenclature isn't too significant, since an elliptic curve has different, and much more interesting, properties as compared to an ellipse.

Instead, an elliptic curve is simply the locus of points in the x-y plane that satisfy an algebraic equation of the form (with some additional minor technical conditions). This is deliberately vague as to what sort of values x and y represent. In the most elementary case, they are real numbers, in which case the elliptic curve is easily graphed in the usual Cartesian plane. But the theory is much richer when x and y may be any complex numbers (in C). And for arithmetic purposes, x and y may lie in some other field, such as the rational numbers Q or a finite field F.

So an elliptic curve is an object that is easily definable with simple high school algebra. Its amazing fruitfulness as an object of investigation may well depend on this simplicity, which makes possible the study of a number of much more sophisticated mathematical objects that can be defined in terms of elliptic curves.

It is very natural to work with curves in the complex numbers, since C is the algebraic closure of the real numbers. That is, it is the smallest algebraically closed field that contains the roots of all possible polynominals with coefficients in R. Being algebraically closed means that C contains the roots of all polynomials with coefficients in C itself. It's natural to work with a curve in an algebraically closed field, since then the curve is as "full" as possible.

The case of elliptic curves in the complex numbers is especially interesting, not only because of the algebraic completeness of C, but also because of the rich analytic theory that exists for complex functions. In particular, the equation of an elliptic curve defines y as an "algebraic function" of x. For every algebraic function, it is possible to construct a specific surface such that the function is "single-valued" on the surface as a domain of definition. It turns out that an elliptic curve, defined as a locus of points, is also the Riemann surface associated with the algebraic function defined by the equation.

So an elliptic curve is a Riemann surface. In fact, it is of a special type: a compact Riemann surface of genus 1. And not only that, but the converse is also true: every compact Riemann surface of genus 1 is an elliptic curve. In other words, elliptic curves over the complex numbers represent exactly the "simplest" sorts of compact Riemann surfaces with non-zero genus. Topologically, the genus counts the number of "holes" in a surface. A surface with one hole is a torus.

This topological equivalence of an elliptic curve with a torus is actually given by an explicit mapping involving the Weierstrass -function and its first derivative. This mapping is, in effect, a parameterization of the elliptic curve by points in a "fundamental parallelogram" in the complex plane.

The -function was originally studied for its analytic properties, specifically the fact that it is doubly periodic. That is, it is periodic with respect to two distinct complex numbers and (where one isn't a real number multiple of the other), in contrast to an exponential or trigonometric function, which has only one fundamental period.

The periodicity of the -function means that it assumes exactly the same values on corresponding points of opposite sides of the parallelogram which is defined by the origin and the two periods and . Therefore, the -function can still be well-defined even if the two pairs of opposite sides of the parallelogram are identified. But when this identification is made, the parallelogram becomes, topologically, a torus. (Imagine taking a rectangular piece of paper and taping together first one pair of opposite sides. You'd have a cylinder. If the paper were flexible enough so you could tape together the two ends of the cylinder, you'd get a torus.)

The topological space that results from identifying opposite sides of a period parallelogram is called a complex torus. The fundamental periods and that define the parallellogram generate a lattice in C consisting of all sums of integral multiples of and . If L denotes the lattice, then L = Z Z. The complex torus can then be described as C/L. What this all means, therefore, is that a (complex) torus is the "natural" domain of definition of the -function, or any doubly periodic complex function.

Classically, such doubly periodic functions were called elliptic functions, since they occurred in the elliptic integrals which represent the arc length of an ellipse. Elliptic curves got their name in this indirect way.

One of the properties of the -function is that it satisfies the equation

where ' is the first derivative and , are constants. Thus, for any z, setting and shows we have an elliptic curve, and the correspondence is the explicit map from the fundamental parallelogram to the elliptic curve, which itself is embedded in C x C (or, more accurately, in PC x PC, where PC is the projective plane, i. e. the Riemann sphere, i. e. C with one "point at infinity" added.)

The group structure of an elliptic curve

Now the plot thickens. It is remarkable enough, if you think about it, that there is such a tidy mapping between a complex torus and an elliptic curve. Especially so, because there are many other noteworthy analytic properties of elliptic functions that were discovered in the 19th century by Weiestrass and others and which we haven't even mentioned. Many of these properties turn out to have simple interpretations in terms of the geometry of Riemann surfaces, which is quite a deep subject in its own right.

But if that were not enough, it happens that elliptic curves have purely algebraic properties which are quite remarkable too. Most importantly, one can easily define an operation on the points of an elliptic curve that turns the whole curve into an abelian group.

Though the definition of the group law is easy, it isn't especially obvious. The simplest way to see it is to go back to looking at the elliptic curve with a given defining equation over the real projective plane, i. e. the ordinary real x-y plane with a point at infinity added. Since the defining equation is a cubic in x, any straight line not parallel to the y-axis (i. e. a line where x isn't constant) will intersect the curve in either 1 point or 3 points. (Since under a rotation this becomes a question about the roots of a cubic polynomial, and there are always either 1 or 3 real roots.)

The definition of the group operation then becomes "simple". If a and b are two distinct points (i. e. x-y pairs) on the curve then they define a straight line which intersects the curve at two points, hence at a third. Suppose the third point on the line is (x, y). Then the result of the group operation, which we denote as a+b, is defined to be (x, -y). The identity element of the group will be the point at infinity, designated as O, so a+O = O+a = a. (This works out precisely because we defined the group operation not as the third point on the line, but instead as the reflection across the x-axis of the point.)

To define a+a, where we have only one point of the curve involved, we use a line which is tangent at the point. (The definition of elliptic curve excludes certain pathological cases of curves which don't have tangents everywhere.) With this definition, then, it is easy, though a little tedious, to verify that points on an elliptic curve do form an abelian group under the + operation, with O as the identity element.

All of the work to define the group structure is purely algebraic, so it can be done over any field, not just the reals. In particular, it can be done over the complex numbers too. In that case, the elliptic curve is a compact Riemann surface - and the group operation makes it a complex Lie group. Such objects have been studied in a very general setting - it has been a very busy area of mathematics for over a hundred years.

The mathematical field of algebraic geometry deals with "curves" in any number of dimensions, called algebraic varieties. When these varieties also have a group operation (that is regular as a mapping of varieties), it is called an algebraic group. It turns out that there are just two kinds of algebraic groups, with very different properties. One kind is a type of algebraic variety with the technical property of being "complete", called an abelian variety, since the group operation (it turms out) must be commutative. The other kind is a linear algebraic group, which is (isomorphic to) an algebraic subgroup of a general linear group - i. e. a group of matrices. Further, the only algebraic group that is of both types is the trivial group. An elliptic curve belongs to the abelian variety type of algebraic group.

We're mentioning all this to emphasize that an elliptic curve is just a special case of a much more general class of objects that have been studied quite extensively in the general setting. But a lot of the motivation for this study comes from the remarkable properties of elliptic curves. One might hope that by finding analogous properties of the more general objects it will become possible, eventually, to prove a vast number of interesting and/or useful results, of which Fermat's Last Theorem is just one example.

In case there is any question about the mention of possible "useful" results, it should be noted that the study of elliptic curves has also led to very concrete results about factoring large numbers, which in turn has an awful lot to do with the contemporary science of cryptography.

Arithmetic on elliptic curves

We are interested in "artithmetical" questions, since the ultimate purpose here is to study diophantine equations, i. e. polynomial equations having integral coefficients, and their solutions which are integral. The Fermat equation is the prime example. In general, an elliptic curve has the form , but for considering arithmetical questions, it is natural to restrict our attention to the case where A, B, C, D are all rational. This assumption will usually be in effect when we are considering properties of elliptic curves involving arithmetical questions (as opposed to their more general analytic properties). If all coefficients are rational, the elliptic curve is said to be defined over Q. The all-important Taniyama-Shimura conjecture concerns only elliptic curves defined over Q.

The fact that any elliptic curve (not necessarily defined over Q) has an abelian group structure means that we can learn a lot about it by studying various of its subgroups. For considering arithmetical (i. e. number theoretic) questions, we restrict our attention to curves defined over Q. In that case, there are several interesting subgroups we can consider.

The first is the group of all points on the curve E which have an order that divides m for some particular integer m. That is, m "times" such a point is the identity element. Such points are called "m-division points", and the subgroup they make up is denoted E[m]. The reason for the name is that any point in E[m] generates a cyclic subroup of E (and E[m]) whose order divides m. If the order is actually m, then the points in the cyclic group generated by the point divide E into m segments.

It isn't necessarily the case that the coordinates of a point in E[m] have integral or rational coordinates. However, the coordinates will be algebraic numbers (i. e., roots of an algebraic equation with coefficients in Q). It's relatively easy to show that as an abstract group E[m] is just the direct sum of two cyclic groups of order m, i. e. Z/mZ Z/mZ, so its order is m. We shall see later that its real interest lies in the fact that we can construct representations of other groups of transformations that act on E[m]. Such representations will consist of 2x2 matrices with integral entries, i. e. elements of GL(Z).

Another interesting subgroup of E is the set of all points whose coordinates are rational. Such points are said to be rational points. If the curve is defined over Q, then it is a simple fact that the set of all rational points (if there are any) is a subgroup.

Many years ago (1921), Louis Mordell proved the theorem named after him, that the group of all rational points on an elliptic curve (over Q) is finitely generated. (There is also a conjecture due to Mordell, that the set of rational points on a algebraic curve of genus > 1 is actually finite. Note that an elliptic curve has genus 1. This conjecture was proved by Gerd Faltings in 1983.)

One of the pricipal facts of elementary group theory is that any finitely generated abelian group is the direct sum of a finite group and a finite number of infinite cyclic groups (isomorphic to the integers Z). If we denote the group of rational points of E by E(Q), then E(Q) = Z E(Q). The number of copies of Z is called the rank of E(Q), and it's a very important invariant of E.

The subgroup E(Q) of torsion points of E(Q), i. e. points of finite order, is also very interesting. A recent (and difficult) theorem by Barry Mazur says that E(Q) must be one of only 15 possible cases, and in fact the order of an element of E(Q) must be 12 (and not 11). There are still many open questions about E(Q), such as how large the rank r can be and whether there is an effective algorithm for computing r. Then there is the famous (among mathematicians) conjecture of Birch and Swinnerton-Dyer which says that r is actually the order of the zero at s=1 of L(E,s), the "L-function" of E, about which function we shall say quite a bit more later. In particular, if this is true, E has infinitely many rational points if and only if L(E,1) = 0. So fans of elliptic curves would like to know a whole lot more about L(E,s).

The definition of L(E,s) will be made based on details about a series of other groups connected with E. These arise by considering E as an elliptic curve over the finite fields F. This is the same as taking the original equation and reducing the coefficients mod p. If the equation of E has rational but non-integral coefficients, we would need to assume none of their denominators are divisible by p, so we might as well assume all coefficients to be integral to begin with (since if the denomimators are prime to p they have inverses mod p). Further, the definition of an elliptic curve requires that there are no repeated roots of the polynomial in x, and this may fail to be true when reducing mod p for some primes. Such primes are said to have "bad reduction". There will be only a finite number of these for any particular curve (they will divide the discriminant), but they have to be dealt with specially.

For any prime p where E has good reduction, we can consider the elliptic curve E(F) over F Since F is finite, there are only a finite number of points on E(F), so it is a finite group. The order of this group, #(E(F)), turns out to be a very important number.

There is a general approach in number theory of trying to deal with "global" problems, such investigating the structure of E(Q), by looking at a closely related "local" problem mod p for all primes p. This is why we are interested in E(F). In particular, if E(F) is "large" for most p, we would expect E(Q) to be large too.

We will see that the numbers #(E(F)) are studied by relating them to coefficients of the Dirichlet series of L(E,s), the L-function of E.

Further basic concepts and results

So far, we have thought of an elliptic curve as defined by its equation. However, a specific equation is not unique for determining a locus of points. Simple substitutions such as x' = 2x obviously lead to different equations with the same locus of points that satisfy the equation. Substitutions like this amount to a change of coordinate system.

It turns out that if an elliptic curve with an equation of the general form y = f(x), where f(x) is a cubic polynomial, then by a change of coordinates, we can find a new equation for the same curve in the form

and this is called the Weierstrass normal form.

In order for this equation to define an elliptic curve, it must have no repeated roots. Elementary algebra shows this happens if and only if the discriminant of x + ax + b, which is 4a + 27b, is not zero. It is customary to define a slightly different form of this:

We say that two elliptic curves are isomorphic if they have defining equations which are the same under some change of coordinate system. Since we can always change coordinates to put the equation in the normal form, we only need to work with that form. However, that form still isn't quite unique - there are different equations in normal form that define isomorphic elliptic curves. In other words, there are coordinate transformations that change the coefficients but preserve the normal form. Such transformations thus lead to isomorphic curves which have different discriminants.

However, it turns out that the quantity

is invariant no matter what normal form of the equation is used. This is called the j-invariant of the elliptic curve. Two elliptic curves are isomorphic if and only if they have the same j-invariant. (The reason for the constant coefficient 1728 is that j, being dependent on the lattice periods and , has an explicit formula in terms of them out of which 1728 falls out naturally.)

Although the discriminant of a defining polynomial isn't an invariant of an elliptic curve, it is close. It happens that there is a related quantity called the minimal discriminant that is invariant. If we consider all equations in normal form for the same elliptic curve, we can choose the one whose discriminant has the fewest distinct prime factors. That discriminant is the minimal discriminant.

The most important fact about the minimal discriminant is that the primes which divide it are precisely the ones at which the curve has bad reduction. In other words, except for those primes, the reduced curve is an elliptic curve over F.

There is still another invariant of an elliptic curve E, called its conductor, and often denoted simply by N. The exact definition is rather technical, but basically the conductor is, like the minimal discriminant, a product of primes at which the curve has bad reduction. Recall that E has bad reduction when it has a singularity modulo p. The type of singularity determines the power of p that occurs in the conductor. If the singularity is a "node", corresponding to a double root of the polynomial, the curve is said to have "multiplicative reduction" and p occurs to the first power in the conductor. If the singularity is a "cusp", corresponding to a triple root, E is said to have "additive reduction", and p occurs in the conductor with a power of 2 or more.

If the conductor of E is N, then it will turn out that N is the "level" of certain functions called modular forms (not yet defined) with which, according to the Taniyama-Shimura conjecture, E is intimately connected.

If N is square-free, then all cases of bad reduction are of the multiplicative type. An elliptic curve of this sort is called semistable. It is for elliptic curves of this sort that Wiles proved the Taniyama-Shimura conjecture.

We might add a few more words about the j-invariant. It is a complex number that characterizes elliptic curves up to isomorphism: two curves are isomorphic if and only if they have the same j-invariant. Not only that, but for any non-zero complex value, there actually exists an elliptic curve with a j-invariant equal to that value. So there is a 1-1 correspondence between (isomorphism classes of) elliptic curves and C*.

Now, we have already seen that an elliptic curve as a complex torus is essentially determined by the period lattice of the -function that parameterizes the curve. More precisely, two tori are isomorphic if and only if their corresponding lattices are "similar", that is, if and only if one is obtained from the other by a "homothety", i. e. multiplication by a non-zero complex number.

But there is another way to characterize similar lattices. Suppose we have two lattices. Each has a Z-basis of the form { , }. Applying a homothety, we can just consider the period ratios and assume the two bases are {1, }, {1, '}, with both and ' in the upper half plane H = {z | Im(z) > 0}. These define the same lattice if and only if they are related by a transformation in SL(Z). This latter is essentially what is known as the modular group . So there is a 1:1 correspondence of similar lattices and elements of H/.

In summary, there are 1:1 correspondences between each of the following

Returning to the j-invariant, it is the 1:1 map betweem isomorphism classes of elliptic curves and C*. But by the above it can also be viewed as a 1:1 map j:H/ -> C. j is therefore an example of what is called a modular function. We'll see a lot more of modular functions and the modular group. These facts, which have been known for a long time, are the first hints of the deep relationship between elliptic curves and modular functions.


Back to Fermat's Last Theorem Home Page

Copyright © 1996 by Charles Daney, All Rights Reserved

Last updated: March 28, 1996